I Einleitung

Dieses Dokument demonstriert einen umfassenden Workflow für Zeitreihenprognosen durch die Anreicherung von Verkaufsdaten mit Feiertagsereignissen. Der Prozess umfasst:

I.a Setup

Erforderlichen Pakete laden.

library(tidyverse)
library(tidymodels)
library(lubridate)
library(timetk)
library(modeltime)
library(prophet)

1 Datenaufbereitung

Zunächst laden wir die Feiertagsdaten und führen sie mit den Verkaufsdaten zusammen. Das holiday_flag wird erstellt, um anzuzeigen, ob ein Datum einem Feiertagsereignis entspricht.

### Load the holidays events data
holidays_events <- read_csv("data/holidays_events.csv")
## Rows: 350 Columns: 6
## ── Column specification ───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
## Delimiter: ","
## chr  (4): type, locale, locale_name, description
## lgl  (1): transferred
## date (1): date
## 
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
party_df <- read_csv("data/Party.csv")
## Rows: 1684 Columns: 6
## ── Column specification ───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
## Delimiter: ","
## chr  (1): family
## dbl  (4): id, store_nbr, sales, onpromotion
## date (1): date
## 
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
party_holidays_promo <- party_df %>%
  left_join(holidays_events, by = "date") %>%
  mutate(holiday_flag = ifelse(is.na(type), 0, 1))

2 Datenaufteilung

Anschließend teilen wir die Daten mithilfe einer Zeitreihenaufteilung in Trainings- und Testdatensätze auf. Der Testdatensatz ist so eingestellt, dass er die Daten eines Jahres auswertet und gleichzeitig einen kumulativen Trainingsdatensatz beibehält.

splits <- time_series_split(
  party_holidays_promo,
  assess = "1 year",
  cumulative = TRUE
)
## Using date_var: date
## Overlapping Timestamps Detected. Processing overlapping time series together using sliding windows.

3 Modelltraining

Wir werden drei verschiedene Modelle trainieren: Prophet, Prophet Boost und ein lineares Regressionsmodell (LM).

3.1 Prophet Model

prophet_model <- prophet_reg() %>%
  set_engine("prophet") %>%
  fit(
    sales ~ date + onpromotion + holiday_flag,
    data = training(splits)
  )
## Disabling daily seasonality. Run prophet with daily.seasonality=TRUE to override this.

3.2 Prophet Boost Model

prophet_boost_model <- prophet_boost() %>%
  set_engine("prophet_xgboost") %>%
  fit(
    sales ~ date + as.numeric(date) + month(date, label = TRUE) + onpromotion + holiday_flag, 
    data = training(splits)
  )
## Disabling daily seasonality. Run prophet with daily.seasonality=TRUE to override this.

3.3 Linear Regression Model

lm_model <- linear_reg() %>%
  set_engine("lm") %>%
  fit(
    sales ~ as.numeric(date) + month(date, label = TRUE) + onpromotion + holiday_flag,
    data = training(splits)
  )

4 Modellzeittabelle und Kalibrierung

Die Modelle werden dann in einer Modell-Zeit-Tabelle kombiniert und anhand der Testdaten kalibriert.

model_tbl <- modeltime_table(
  prophet_model,
  prophet_boost_model,
  lm_model
)

calibration_tbl <- model_tbl %>%
  modeltime_calibrate(testing(splits))

Modellleistung anhand der Genauigkeitstabelle überprüfen:

calibration_tbl %>%
  modeltime_accuracy() %>%
  table_modeltime_accuracy(resizable = TRUE, bordered = TRUE)
## ℹ We have detected a possible intermittent series, you can change the default metric set to the extended_forecast_accuracy_metric_set() containing the MAAPE metric, which is more appropriate for this type of series.

5 Forecasting

Prognostizieren Sie die zukünftigen Werte mit den kalibrierten Modellen und visualisieren Sie die Ergebnisse.

5.1 Prognose auf Grundlage von Testdaten

calibration_tbl %>%
  modeltime_forecast(
    new_data = testing(splits),
    actual_data = party_holidays_promo,
    conf_interval = 0.95
  ) %>%
  plot_modeltime_forecast(.legend_show = TRUE, .legend_max_width = 25)

5.2 Modelle auf allen Daten und Prognosen neu anpassen

Kalibrierten Modelle mithilfe des gesamten Datensatzes und die Prognosen auf Grundlage der Testdaten:

refit_tbl <- calibration_tbl %>%
  modeltime_refit(data = party_holidays_promo)
## Disabling daily seasonality. Run prophet with daily.seasonality=TRUE to override this.
## Disabling daily seasonality. Run prophet with daily.seasonality=TRUE to override this.
forecast_tbl <- refit_tbl %>%
  modeltime_forecast(
    new_data = testing(splits),
    actual_data = party_holidays_promo,
    conf_interval = 0.95
  )

### Plot interactive forecast if supported by your output&emsp;device
forecast_tbl %>%
  plot_modeltime_forecast(.interactive = TRUE)

6 Modell-Mittelwertbildung

Hier führen wir eine Modellmittelung durch, indem wir die Prognosen (ohne die tatsächlichen Werte) gruppieren und die mittlere Vorhersage zusammen mit Konfidenzintervallen berechnen.

mean_forcast_tbl <- forecast_tbl %>%
  filter(.key != "actual") %>%
  group_by(.key, .index) %>%  # Group all predictions for one date
  summarise(across(.value:.conf_hi, mean)) %>%  # Compute the mean for each forecast metric
  mutate(
    .model_id = 3,     # Denotes the number of models used
    .model_desc = "Average of Models"
  )
## `summarise()` has grouped output by '.key'. You can override using the
## `.groups` argument.
### Combine actual values with the averaged forecasts and plot
forecast_tbl %>%
  filter(.key == "actual") %>%
  bind_rows(mean_forcast_tbl) %>%
  plot_modeltime_forecast()

7.1 Zugriff auf die Prognosewerte

Überprüfen Sie die numerischen Prognosewerte

forecast_tbl$.value
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## [1177]   7.000000  15.000000  12.000000  14.000000   4.000000  15.000000
## [1183]  30.000000  15.000000  23.000000  26.000000  18.000000  14.000000
## [1189]   9.000000  11.000000   8.000000  21.000000  24.000000  13.000000
## [1195]  16.000000  11.000000   5.000000   9.000000  22.000000  44.000000
## [1201]  19.000000  21.000000  14.000000   9.000000  15.000000  28.000000
## [1207]  47.000000  17.000000  14.000000  26.000000  11.000000  20.000000
## [1213]  16.000000  17.000000  18.000000  12.000000  14.000000  10.000000
## [1219]  28.000000  28.000000  16.000000  35.000000  17.000000  10.000000
## [1225]   7.000000   9.000000   6.000000  13.000000  34.000000  18.000000
## [1231]  18.000000  13.000000  23.000000   4.000000  20.000000  31.000000
## [1237]  40.000000  40.000000   9.000000   9.000000   4.000000   5.000000
## [1243]  12.000000  20.000000  20.000000  20.000000  47.000000  10.000000
## [1249]  13.000000  17.000000  12.000000   3.000000 164.000000  46.000000
## [1255]  14.000000  10.000000  19.000000   5.000000  14.000000  29.000000
## [1261]  32.000000  13.000000   7.000000  13.000000   9.000000  21.000000
## [1267]  18.000000  28.000000   7.000000  10.000000   7.000000   8.000000
## [1273]  19.000000  25.000000  34.000000   7.000000   3.000000   8.000000
## [1279]  24.000000   9.000000  21.000000  25.000000   6.000000   9.000000
## [1285]  31.000000   7.000000  18.000000  12.000000  25.000000  25.000000
## [1291]  25.000000  32.000000  13.000000  16.000000  12.000000  15.000000
## [1297]  23.000000  34.000000  15.000000  15.000000  10.000000  12.000000
## [1303]  19.000000  25.000000   7.000000  18.000000   4.000000   4.000000
## [1309]   5.000000   4.000000   4.000000   5.000000  12.000000   6.000000
## [1315]   4.000000   5.000000   3.000000   8.000000  15.000000  18.000000
## [1321]  16.000000  16.000000   5.000000  11.000000  13.000000   7.000000
## [1327]  21.000000  39.000000  10.000000  14.000000   1.000000   8.000000
## [1333]  10.000000 135.000000  13.000000   8.000000   4.000000  15.000000
## [1339]  22.000000  22.000000  14.000000  19.000000   7.000000   7.000000
## [1345]  10.000000  29.000000   5.000000  21.000000  53.000000  20.000000
## [1351]  13.000000   6.000000  14.000000  34.000000  23.000000  38.000000
## [1357]  11.000000  13.000000  28.000000   9.000000  21.000000  15.000000
## [1363]  12.000000   8.000000   4.000000   6.000000   2.000000   3.000000
## [1369]  14.000000  35.000000  10.000000   0.000000   7.000000   9.000000
## [1375]  22.000000  21.000000  21.000000  14.000000  13.000000  16.000000
## [1381]   8.000000  10.000000  13.000000  30.000000  18.000000   7.000000
## [1387]   3.000000   9.000000   2.000000  10.000000  25.000000   9.000000
## [1393]   6.000000   7.000000  14.000000  14.000000  20.000000  29.000000
## [1399]  18.000000   0.000000   6.000000   7.000000  11.000000   6.000000
## [1405]  15.000000   7.000000   8.000000   5.000000   3.000000  16.000000
## [1411]   8.000000   4.000000   0.000000   1.000000   3.000000   0.000000
## [1417]   8.000000  14.000000  18.000000   9.000000  25.000000   3.000000
## [1423]   1.000000   3.000000   9.000000  11.000000   5.000000   7.000000
## [1429]   7.000000   1.000000  10.000000   7.000000  13.000000  13.000000
## [1435]   6.000000   6.000000   3.000000   4.000000   1.000000   0.000000
## [1441]   4.000000   7.000000   1.000000   6.000000   5.000000   5.000000
## [1447]   6.000000  21.000000   5.000000   1.000000   4.000000   1.000000
## [1453]   8.000000   6.000000  21.000000   2.000000   5.000000   8.000000
## [1459]   5.000000   8.000000  12.000000  11.000000   3.000000   2.000000
## [1465]  11.000000   6.000000  25.000000  12.000000  18.000000  18.000000
## [1471]   3.000000  11.000000  62.000000  55.000000  55.000000  27.000000
## [1477]  34.000000  10.000000  16.000000  23.000000  13.000000   6.000000
## [1483]  17.000000   0.000000  11.000000  22.000000  14.000000  14.000000
## [1489]   8.000000  36.000000  18.000000   5.000000  16.000000  27.000000
## [1495]  30.000000   9.000000  42.000000  26.000000  11.000000  14.000000
## [1501]  12.000000  11.000000  12.000000  43.000000   5.000000   2.000000
## [1507]  14.000000  15.000000  10.000000  50.000000  21.000000  11.000000
## [1513]   4.000000  21.000000 152.000000  11.000000  27.000000  15.000000
## [1519]   8.000000   8.000000  11.000000  15.000000   9.000000  11.000000
## [1525]  32.000000   8.000000   8.000000   8.000000   7.000000  13.000000
## [1531]  10.000000  20.000000   6.000000   8.000000  13.000000   8.000000
## [1537]  12.000000  25.000000  22.000000  12.000000  12.000000  15.000000
## [1543]   7.000000  15.000000  11.000000  16.000000  14.000000  22.000000
## [1549]  10.000000   9.000000  11.000000   8.000000  15.000000   6.000000
## [1555]   8.000000  10.000000  16.000000  22.000000  20.000000  24.000000
## [1561]  20.000000   3.000000   6.000000   5.000000   9.000000  10.000000
## [1567]  11.000000  13.000000   3.000000   6.000000   8.000000   6.000000
## [1573]  14.000000  13.000000   4.000000   4.000000   5.000000  13.000000
## [1579]  13.000000  19.000000  20.000000   7.000000   8.000000   8.000000
## [1585]   6.000000  17.000000  11.000000  11.000000  12.000000   8.000000
## [1591]   7.000000   1.000000   4.000000  11.000000   6.000000   6.000000
## [1597]   8.000000  15.000000  13.000000   9.000000   5.000000   3.000000
## [1603]  14.000000   7.000000   6.000000   7.000000   5.000000   6.000000
## [1609]   8.000000  23.000000  18.000000  27.000000   7.000000   6.000000
## [1615]  11.000000   8.000000  15.000000  22.000000   7.000000   7.000000
## [1621]   7.000000  12.000000  13.000000  27.000000  24.000000  12.000000
## [1627]   8.000000  38.000000  11.000000  23.000000  22.000000  14.000000
## [1633]   7.000000   5.000000   7.000000  10.000000  16.000000  28.000000
## [1639]  13.000000   5.000000   3.000000   7.000000 108.000000   0.000000
## [1645]  23.000000  16.000000  10.000000  17.000000  10.000000   7.000000
## [1651]  11.000000  29.000000  13.000000  16.000000  11.000000   9.000000
## [1657]   7.000000  19.000000  49.000000  13.000000  13.000000  13.000000
## [1663]   6.000000  10.000000  28.000000  18.000000  32.000000  21.000000
## [1669]  12.000000  10.000000  10.000000  11.000000  14.000000  18.000000
## [1675]  35.000000  43.000000  20.000000  14.000000  18.000000  14.000000
## [1681]  10.000000   8.000000  34.000000  23.000000   7.000000  20.000000
## [1687]  17.000000   7.000000  14.000000  31.000000  19.000000  11.000000
## [1693]  18.000000  12.000000  10.000000  18.000000  32.000000  13.000000
## [1699]  11.000000  20.000000   9.000000   5.000000  11.000000  25.000000
## [1705]  25.000000  11.000000  18.000000  20.000000  16.000000  10.000000
## [1711]  18.000000  11.000000  12.000000  16.000000  12.202093  12.672475
## [1717]  13.999122  16.407220  21.009409  13.831997  10.388916  12.338808
## [1723]  14.211520  14.265210  16.729410  21.381455  14.247531  10.841668
## [1729]  12.822814  13.386349  14.795993  17.277184  21.943124  14.820959
## [1735]  11.425693  13.417212  13.991760  15.413795  17.909327  22.591936
## [1741]  15.488913  12.138931  14.177735  14.800938  16.272105  18.816114
## [1747]  23.545196  16.485116  13.149117  15.194978  15.816551  17.275680
## [1753]  19.795631  24.487189  17.374913  13.971125  15.933036  17.789284
## [1759]  17.797064  20.185235  24.730777  17.459866  13.886888  15.671528
## [1765]  16.010306  17.168472  20.708359  23.740588  17.634850  12.569310
## [1771]  14.212082  14.428046  15.485481  17.614392  21.932457  14.471120
## [1777]  10.749413  12.430906  12.715108  13.869261  16.121639  20.587518
## [1783]  13.295308   9.760440  11.642385  12.136151  13.504183  19.308714
## [1789]  25.651555  17.721396  11.028872  12.237393  14.203414  15.681211
## [1795]  18.226178  21.611444  15.863984  12.489033  13.141934  13.684801
## [1801]  16.378640  18.776818  23.326004  23.326004  14.717583  11.136441
## [1807]  12.908224  13.230696  14.370613  16.557054  20.907386  13.453357
## [1813]   9.714917  11.357140  11.581599  12.658110  16.152939  19.179884
## [1819]  11.775981   9.460620   9.892483  10.275634  11.541674  13.917887
## [1825]  18.518282  11.369767   9.321052  11.359861  10.679323  13.544891
## [1831]  14.838164  19.672559  12.735003   9.534822  11.725969  12.498668
## [1837]  14.111445  16.784648  21.626628  14.660044  11.395900  13.490684
## [1843]  15.472902  16.935156  16.935156  19.437335  24.092812  13.464095
## [1849]  14.023392  14.476272  15.755347  18.090159  23.933108  16.645648
## [1855]  13.081878  13.570635  13.981692  15.247743  17.597530  22.146835
## [1861]  14.924594  11.446771  13.373370  13.899926  15.289403  17.765605
## [1867]  22.439272  15.334350  11.962007  13.977785  14.573231  16.007905
## [1873]  18.502899  23.167016  16.023100  12.582091  14.500193  14.970488
## [1879]  16.254916  18.577731  23.051599  15.703825  12.050295  13.752507
## [1885]  14.009049  15.087424  17.217359  21.516757  14.017777  10.240644
## [1891]  11.850564  12.049095  13.105892  15.251953  19.605532  12.198209
## [1897]   8.548702  10.319830  10.709928  11.984683  14.370473  18.980397
## [1903]  11.840476   8.463606  10.506658  11.162052  12.689732  15.310649
## [1909]  20.132921  13.178152  11.290454  13.453773  12.858350  15.767705
## [1915]  17.062487  21.856146  14.836604  11.515095  13.548316  14.129975
## [1921]  15.522714  17.951489  22.529614  15.284917  11.733626  13.537352
## [1927]  13.894517  15.072161  17.299216  21.692463  14.282612  10.588133
## [1933]  12.272204  12.534123  13.641117  15.821640  20.191375  12.779366
## [1939]   9.101929  10.819682  11.129056  12.294198  14.540388  20.315097
## [1945]  11.639408   8.031668   9.814958  10.183504  11.399685  13.687487
## [1951]  18.158636  10.838859   7.241877   9.026512  10.723197  11.926038
## [1957]  14.196000  14.196000  17.312131   9.972569   6.358654   8.131801
## [1963]   8.489989   9.699588  13.323638  16.472994   9.182094   5.633154
## [1969]   7.487966   7.944363   9.268040  11.685870  16.311907   9.173533
## [1975]   7.120329   7.806288   8.430973   9.921771  12.502132  18.117428
## [1981]  10.287527   7.028148   9.160623   9.877174  11.438145  15.400444
## [1987]  20.203817  13.206920   8.586502  10.669359  11.314113  13.617748
## [1993]  15.300393  20.813495  12.844975   9.414213  11.346974  15.679747
## [1999]  13.160980  16.871119  20.082021  12.828597   9.293840  11.140616
## [2005]  11.568118  12.843673  15.196042  19.741386  12.509394   9.017149
## [2011]  10.926123  11.433648  12.804772  15.265628  19.929477  12.822912
## [2017]   9.459807  11.498406  12.132865  13.625199  16.198720  20.964146
## [2023]  13.945868  10.655941  12.751208  13.424870  14.938435  18.850121
## [2029]  22.266131  16.551496  16.551496  16.551496  11.879561  13.913539
## [2035]  14.513406  15.942622  18.426534  23.078417  15.925633  13.818500
## [2041]  13.818500  14.412480  14.909294  16.239117  18.628738  23.192603
## [2047]  15.958964  12.443766  14.307667  14.747512  16.028201  18.376051
## [2053]  22.904837  15.641981  12.102489  13.946009  14.368359  15.633432
## [2059]  17.966610  22.480840  16.537819  12.982101  14.807965  13.876127
## [2065]  15.120479  17.431702  21.923194  14.622305  11.044279  12.849298
## [2071]  13.233985  14.463281  16.763804  22.584849  13.949716  10.379271
## [2077]  12.199311  12.606861  15.202032  17.541512  20.739410  13.493427
## [2083]   9.983935  13.205846  13.644157  16.812144  18.131508  20.591770
## [2089]  25.161081  17.965154  14.507385  16.395215  23.586571  18.279736
## [2095]  20.794749  25.413842  18.262623  14.844588  16.767481  17.304108
## [2101]  13.892474  14.889775  16.937126   9.807446   6.410786   8.355726
## [2107]   8.915812  10.347374  13.858835  18.531057  11.434510   8.094749
## [2113]  10.098304  10.717875  13.943085  16.568195  21.294939  14.248055
## [2119]  10.929790  12.946259  12.518597  13.999573  16.600909  21.288477
## [2125]  14.185836   8.531381   8.813957  11.449093  10.681653  13.140141
## [2131]  19.274353  12.000718   8.427573  10.163412  10.482244  11.641453
## [2137]  15.085920  18.250515  11.979810   6.195996   7.788429   7.985432
## [2143]   9.047516  11.243443  15.546553   8.087791   4.375684   6.022569
## [2149]   6.305283   7.482919   9.822079  14.293224   7.023757   3.518074
## [2155]   5.383875   5.892965   7.299122  13.524078  17.838215   9.511191
## [2161]  18.940428   3.599380   5.418052   6.923290   9.552201  13.088996
## [2167]   7.190967   3.820899   4.582571   5.102905   7.634785  10.082493
## [2173]  14.592110  14.592110   6.110796   2.500816   4.193174   4.463520
## [2179]   5.571172   7.784381  12.076661   6.402663   2.626770   4.729680
## [2185]   4.904316   5.956425   9.335169  13.127823   5.704605   3.223574
## [2191]   3.760844   4.129331   5.395677   7.851479  12.452845   5.325558
## [2197]   3.927560   5.953776   4.673919   8.205809   8.986966  13.846566
## [2203]   6.949476   3.793429   9.154661   9.950762  15.300792  18.072917
## [2209]  22.925133  15.978796  12.733755  14.794793  40.768600  42.237262
## [2215]  42.237262  25.995112  30.621284  15.044465  14.910535  15.316488
## [2221]  14.503020  16.866612  23.272171  15.940244  12.335341  12.891792
## [2227]  13.244922  14.476080  16.852985  24.569091  17.309352  13.801190
## [2233]  15.653500  16.139251  17.513356  20.038042  24.679334  17.560279
## [2239]  14.181129  16.145800  16.722795  18.163199  20.726164  25.375483
## [2245]  18.232878  14.798228  16.676036  17.136348  18.432698  20.827535
## [2251]  25.288707  17.942635  14.294421  15.954163  94.143429  15.174781
## [2257]  21.203031  19.801412  12.300913   9.760590  11.326187  11.510795
## [2263]  11.762241  13.977713  21.522516  11.548692   7.905668   9.636899
## [2269]  10.017933  11.306222  13.766432  18.366852  11.232550   8.505192
## [2275]  10.510643  11.158034  12.698913  18.136067  21.589342  14.635153
## [2281]  12.265397  14.383031  11.064256  13.820070  15.332144  20.100777
## [2287]  13.069412   9.738721  10.228735  10.781639  12.166723  14.647672
## [2293]  19.193487  11.931590   9.352327  11.097356  13.165496  14.339317
## [2299]  16.624786  20.994287  13.577983   6.865969   8.505799   8.756254
## [2305]   9.876402  12.133042  16.497075   9.096605   5.437007   7.127250
## [2311]   7.440718   8.633080  10.967682  17.114372   8.614530   5.030468
## [2317]   6.789756  10.544085  11.786429  14.159479  18.630142  10.307171
## [2323]   6.725712   8.477104  10.012897  11.231507  12.573785  12.573785
## [2329]  15.835438   8.499203   4.892112   6.624443   6.971672   8.192259
## [2335]  10.579401  13.872630   8.014704   6.479019   8.295628   7.378931
## [2341]   7.631225  10.125623  14.744530   7.614379   7.076088   5.976014
## [2347]   6.595775   8.102855  10.758754  22.862370  14.653108  11.398658
## [2353]   8.379323   9.078450  10.640913  15.468408  20.242766  13.228390
## [2359]   7.784764   9.800534  10.405179  13.434265  14.407213  25.530122
## [2365]  13.781342  10.317161  10.690464  31.743996  12.424520  19.419813
## [2371]  19.328460  12.042427   7.712627   9.490016   9.881989  11.397222
## [2377]  15.464620  19.985611  12.748945   6.438462   8.309289   8.813301
## [2383]  65.924107   6.974798  21.832181  14.749929  11.417909  13.442410
## [2389]  11.439045  12.971439  15.645490  22.248777  15.260049  12.002218
## [2395]  12.707067  13.390897  14.933510  20.253309  30.247981  16.174796
## [2401]  16.174796  16.174796  14.060235  16.052613  16.637130  18.069244
## [2407]  20.612935  25.237107  18.068698  13.399805  13.399805  16.478617
## [2413]  16.940877  18.256968  22.560677  27.086027  19.052430  15.518813
## [2419]  17.318206  14.204141  15.475497  17.876335  23.960363  16.685586
## [2425]  13.140765  14.934193  15.338906  16.610566  19.012832  23.513819
## [2431]  17.880481  14.333955  16.123106  14.879393  16.140724  18.529665
## [2437]  23.014665  15.722027  12.156004  14.177511  12.714762  13.958109
## [2443]  16.331698  25.290896  16.350129  12.782792  14.557011  14.947031
## [2449]  16.800462  17.176359  20.907544  13.649934  11.650025  14.256111
## [2455]  14.353161  14.364292  14.375422  14.386552  14.397682  14.408812
## [2461]  14.419943  14.431073  16.583802  14.453333  14.464464  14.475594
## [2467]  14.486724  14.497854  14.508984  14.520115  17.577698  17.588828
## [2473]  17.599959  17.611089  17.622219  17.633349  17.644479  17.655610
## [2479]  17.666740  17.677870  17.689000  17.700131  17.711261  17.722391
## [2485]  17.733521  17.744651  17.755782  17.766912  17.778042  17.789172
## [2491]  17.800302  17.811433  17.822563  17.833693  17.844823  17.855954
## [2497]  17.867084  20.019813  17.889344  17.900474  17.565455  17.576585
## [2503]  17.587715  17.598846  17.609976  17.621106  19.773835  17.643367
## [2509]  19.796096  17.665627  17.676757  17.687887  17.699018  17.710148
## [2515]  17.721278  17.732408  17.743538  17.754669  17.765799  17.776929
## [2521]  17.788059  17.799190  17.810320  17.821450  17.832580  17.843710
## [2527]  17.854841  20.790841  22.264407  21.544320  18.630579  16.157691
## [2533]  18.310420  18.321550  18.332680  16.202211  18.354941  18.366071
## [2539]  16.235602  16.246732  18.399462  18.410592  18.421722  18.421722
## [2545]  16.291253  16.302383  16.313514  16.324644  16.335774  16.346904
## [2551]  16.358034  16.369165  16.380295  16.391425  16.402555  16.413686
## [2557]  18.566415  16.435946  16.447076  18.599805  16.469337  16.480467
## [2563]  16.590927  16.602057  16.613187  16.624317  18.777047  18.788177
## [2569]  16.657708  18.810437  16.679969  16.691099  16.702229  16.713359
## [2575]  16.724489  16.735620  16.746750  16.757880  16.769010  16.780140
## [2581]  16.791271  16.802401  18.955130  18.966260  18.966260  18.977390
## [2587]  18.988521  19.010781  16.880312  16.891443  16.902573  16.913703
## [2593]  19.066432  19.757535  19.768665  17.638196  17.649327  17.660457
## [2599]  17.671587  17.682717  17.693847  17.704978  17.716108  17.727238
## [2605]  17.738368  17.749499  17.760629  17.771759  17.782889  17.794019
## [2611]  17.805150  17.816280  17.827410  17.838540  17.849670  17.860801
## [2617]  17.871931  17.883061  17.894191  17.905322  17.916452  17.927582
## [2623]  17.938712  17.949842  16.037926  16.049056  16.060186  16.071316
## [2629]  16.082447  16.093577  16.104707  16.115837  16.126967  16.138098
## [2635]  16.149228  16.160358  16.171488  16.182619  16.193749  16.204879
## [2641]  16.216009  16.227139  16.238270  16.249400  16.260530  16.271660
## [2647]  16.282790  16.293921  16.305051  16.316181  18.468910  18.480041
## [2653]  17.004368  19.157097  17.026628  17.037759  17.048889  17.060019
## [2659]  17.071149  17.082279  17.093410  17.104540  17.115670  17.126800
## [2665]  17.137930  17.149061  17.160191  17.171321  17.182451  17.193582
## [2671]  17.204712  17.215842  17.226972  17.238102  17.249233  17.260363
## [2677]  17.271493  17.282623  17.293753  17.304884  17.316014  17.327144
## [2683]  17.338274  15.290172  13.159704  13.170834  13.181964  13.193094
## [2689]  13.204225  13.215355  13.226485  13.237615  13.248745  13.259876
## [2695]  15.412605  15.423735  15.434865  15.434865  13.304396  13.315527
## [2701]  13.326657  13.337787  13.348917  13.360048  15.512777  13.382308
## [2707]  13.393438  13.404568  13.415699  13.426829  13.437959  13.449089
## [2713]  13.460219  13.471350  16.874016  14.743547  14.754678  14.765808
## [2719]  14.776938  15.519286  14.799199  14.810329  14.821459  14.832589
## [2725]  14.843719  16.996449  17.007579  17.018709  14.888240  14.899370
## [2731]  14.910501  15.652849  14.932761  15.675109  14.955022  14.966152
## [2737]  14.977282  19.323664  14.999542  17.152272  15.021803  15.032933
## [2743]  15.044063  15.055193  15.066324  17.424263  17.435394  17.446524
## [2749]  17.457654  17.468784  17.479914  17.491045  17.502175  17.513305
## [2755]  17.524435  17.535565  17.546696  17.557826  17.568956  17.580086
## [2761]  17.591217  17.602347  17.613477  17.624607  17.635737  17.646868
## [2767]  17.657998  19.810727  17.680258  19.832987  19.832987  19.832987
## [2773]  17.702519  17.713649  17.724779  17.735909  17.747040  19.728088
## [2779]  19.739218  21.891947  21.891947  19.761479  19.772609  19.783739
## [2785]  19.794869  19.805999  19.817130  19.828260  19.839390  19.850520
## [2791]  19.861651  19.872781  19.883911  19.895041  19.906171  19.917302
## [2797]  19.928432  19.939562  19.950692  19.961822  22.114552  22.125682
## [2803]  22.136812  20.006343  20.017474  20.028604  20.039734  20.050864
## [2809]  20.061994  18.248736  18.259866  18.270997  18.282127  20.434856
## [2815]  18.304387  18.315517  18.326648  18.337778  20.490507  20.501637
## [2821]  18.371169  18.382299  18.393429  20.546158